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Overview of the Numerical Methods for the Modelling of Rock Mechanics Problems

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Overview of the Numerical Methods for the Modelling of Rock Mechanics Problems M. Nikolić i dr. Pregled numeričkih metoda za modeliranje u mehanici stijena ISSN 1330-3651 (Print), ISSN 1848-6339 (Online) DOI: 10.17559/TV-20140521084228 OVERVIEW OF THE NUMERICAL METHODS FOR THE MODELLING OF ROCK MECHANICS PROBLEMS Mijo Nikolić, Tanja Roje-Bonacci, Adnan Ibrahimbegović Subject review The numerical methods have their origin in the early 1960s and even at that time it was noted that numerical methods can be successfully applied in various engineering and scientific fields, including the rock mechanics. Moreover, the rapid development of computers was a necessary background for solving computationally more demanding problems and the development process of the methods in general. Thus, we have many different methods presently, which can be separated into two main branches: continuum and discontinuum-based numerical methods. Some problems require the strengths of both main approaches which brought the hybrid continuum/discontinuum methods. The first goal of this paper is to present the state of the art numerical methods and approaches for solving the rock mechanics problems, as well as to give the brief explanation about the theoretical background of each method. The second goal is to emphasise the area of applicability of the methods in rock mechanics. Keywords: numerical methods; numerical modelling; rock mechanics Pregled numeričkih metoda za modeliranje u mehanici stijena Pregledni rad Počeci numeričkih metoda sežu u rane 1960-e. Već tada je bilo jasno da numeričke metode mogu biti uspješno upotrijebljene za različita inženjerska i znanstvena područja, uključujući i primjenu u mehanici stijena. Ubrzan razvoj računala je omogućavao razvoj numeričkih metoda i rješavanje računalno zahtjevnijih sustava. Takav razvoj doveo je do velikog broja različitih metoda i pristupa koji se mogu svrstati u dvije skupine: metode kontinuuma i metode diskontinuuma. Određene zadaće zahtijevaju prednosti oba pristupa, što je dovelo do razvoja kombiniranih konačno-diskretnih metoda. Prvi cilj ovog rada je predstavljanje numeričkih metoda i pristupa koji se koriste za rješavanje zadaća u mehanici stijena, kao i kratko objašnjenje osnovnih teorijskih postavki svake od metoda. Drugi cilj je osvrt na primjenjivost pojedine metode u mehanici stijena. Ključne riječi: numeričke metode; numeričko modeliranje; mehanika stijena 1 Introduction development of computers made a significant contribution to the field called computational mechanics which found Rock material is a natural geological material its wide use in rock mechanics. Namely, the numerical consisting of different minerals. It is discontinuous, methods for approximately solving the partial differential anisotropic, inhomogeneous, inelastic and contains equations are used today as the main approach in numerous randomly oriented zones of initiation of engineering design and research of rock mechanics. In potential failure. These zones can be initial cracks, this paper we are presenting the state of the art of defects, cavities or other natural flaws. Rocks have been numerical methods in terms of historical background as used (by humans) from the early beginnings of the human well as their strengths and weaknesses with the focus on race in many different ways especially for the building deterministic approach. purposes. We were constantly facing its unpredictable Numerical model for the analysis of rock mass nature trying to build roads, tunnels, underground problems has to include the possibility of description of excavations, mining shafts or avoid the sudden rock falls the rock behaviour by relevant material model and and slopes. From the early days there was a need for fracture mechanism, the presence of the pre-existing predicting the behaviour of this material and up to present cracks, the pre-existing stress state, inhomogeneity and times, we are still trying to fully understand it. anisotropy as well as a time dependent behaviour caused Because of the irregular nature of rock material, the by creep and plastic deformation. The additional prediction of its behaviour has always been a challenging challenge is coupling of the hydraulic and thermal process task. Rock mechanics developed mostly for the design of in analysing the mechanical behaviour of the rock. The rock engineering structures and today we have a wide choice of the appropriate model and numerical methods spectrum of different modelling approaches for many depends on the problem which has to be solved. various rock mechanics problems. That includes the The numerical methods can be classified into three approaches based on the previous experiences, simplified main categories: continuum, discontinuum and hybrid mathematical models that can be solved analytically (e.g. continuum/discontinuum methods. Namely, the Bishop Method of slices for slope stability) or general continuum concept implies that the domain of interest classification systems for rocks. The latter found large cannot be separated and the continuity between the points number of applications in various types of engineering must be preserved in order to establish the derivatives. projects like designing and construction of excavations in Thus the continuity between the elements in numerical rock. There are several rock mass classification systems approach must be preserved as well. Contrary to the including RMR, Q and GSI classifications which are used analytical solution of differential equations where the to estimate the rock mass properties. The lack of solution is known in each point, the numerical solution is information is a common fact in rock mechanics and calculated in the finite number of pre-defined nodes engineering design so empirical approaches (including the which reduces the system complexity. The discontinuum classification systems) are also widely used. The approach on the other side treats the separate elements as Tehnički vjesnik 23, 2(2016), 627-637 627 Overview of the numerical methods for the modelling of rock mechanics problems M. Nikolić et al. discrete ones which are individually continuous and they at the surrounding nodes. After introducing the boundary interact between each other. In the discontinuous conditions, the system of algebraic equations is finally methods, the rigid body motion (usually with large solved (usually by direct or iterative methods) which movements) is the main case of interest, while in produces the values of unknowns at each node leading to continuum-based methods the deformation of the system the approximate solution of the partial differential is the main objective. Thus, the sliding of the rock block equation with an error made because of the difference in on the pre-existing discontinuity would rather be partial derivatives and finite differences. Such a direct computed with the discrete methods, while the choice of kind of discretization, together with no practical need to continuum method would be better used in calculating the use interpolation functions as in other methods like FEM deformation of the rock mass above the excavation of the or BEM, position this method as the most direct and tunnel. In addition there also exist the hybrid intuitive technique for solving the partial differential continuum/discontinuum methods that use the best equations. The advantage of the method is also in properties of both approaches. Such an example is to have possibility of the simulation of complex nonlinear the discrete rock block which moves and also deforms material behaviour without iterative solutions. under the external loading. The standard FDM uses regular grid, such as rectangular which is also the most important shortcoming Table 1 Overview of the most notable numerical approaches and of the method. Thus, representing the irregular geometries methods together with dealing with heterogeneous nature of rocks Continuum Methods: and complex boundary conditions seems like a significant ▪Finite Difference Method FDM ▪Finite Volume Method FVM disadvantage of this method for simulating rock ▪Finite Element Method FEM behaviour in practical rock mechanics tasks. Another ▪Meshless Methods important disadvantage lies in continuity requirement of ▪Boundary Element Methods BEM the proposed equations which is not suitable for dealing Discontinuum Methods: with fractures. However, the general FDM method [1, 2] ▪Discrete Element Method DEM evolved eventually to deal with these shortcomings, ▪Discrete Fracture Network Method DFN Hybrid Methods: mostly with irregular grids, which include irregular ▪Discrete Finite Element Method quadrilateral, triangular and Voronoi grids. ▪Combined Finite Discrete Element Method FEM/DEM Further development of this approach continued with the Finite Volume Method (FVM) [3]. The FVM is also a The most notable continuum numerical methods are method for solving partial differential equations, not Finite Difference Method (FDM), Finite Volume Method directly as in FDM, but in integral sense. This leads to the (FVM), Finite Element Method (FEM), Meshless formulation of finite volumes, which represent the volume Methods and Boundary Element Method (BEM). The surrounding each node in mesh. The basic principle is to discontinuum methods are Discrete Element Method replace the integrals with algebraic functions of the (DEM) and Discrete Fracture Network (DFN). The most unknowns in the nodes, taking into account the initial and notablehybrid continuum/discontinuum methods with boundary conditions which lead to the set of algebraic applications in rock mechanics are Discrete Finite
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